Question

Racionalize o Denominador dos Números a Seguir.

Ps: Letras A e C & G já foram respondidas.

A)

$\frac{1}{ \sqrt{3} } = \frac{1 \sqrt{3} }{3}$

B)

$\frac{3}{ 2\sqrt{5} }$

C)

$\frac{2}{ \sqrt{8} } = \frac{2 \sqrt{8} }{8}$

D)
$\frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{3} }$

E)
$\frac{3 + \sqrt{3} }{ \sqrt{3} }$

F)
$\frac{10}{2 - \sqrt{2} }$

G)
$\frac{3}{ \sqrt{5} + \sqrt{3} } = \frac{3 \sqrt{5} - 3 \sqrt{3} }{2}$

H)
$\frac{ \sqrt{2} }{ \sqrt{3} - 1}$

I)
$\frac{ \sqrt{11} + 1}{ \sqrt{11} - 1 }$

J)
$\frac{7}{2 + \sqrt{5} }$

​

Answer 1

Explicação passo-a-passo:

A)

$\dfrac{1}{ \sqrt{3} }$

$\dfrac{1}{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\dfrac{1 \sqrt{3} }{ \sqrt{3} \sqrt{3} }$

$\dfrac{ \sqrt{3} }{ \sqrt{9} }$

$\dfrac{ \sqrt{3} }{3}$

B)

$\dfrac{3}{2 \sqrt{5} }$

$\dfrac{3}{2 \sqrt{5} } \times \dfrac{ \sqrt{5} }{ \sqrt{5} }$

$\dfrac{ 3\sqrt{5} }{2 \sqrt{5} \sqrt{5} }$

$\dfrac{3 \sqrt{5} }{2 \sqrt{25} }$

$\dfrac{3 \sqrt{5} }{2 \times 5}$

$\dfrac{3 \sqrt{5} }{10}$

C)

$\dfrac{2 }{ \sqrt{8} }$

$\dfrac{2}{ \sqrt{2 {}^{3} } }$

$\dfrac{2}{ \sqrt{2 {}^{2 + 1} } }$

$\dfrac{2}{ \sqrt{2 {}^{2} \times 2 {}^{1} } }$

$\dfrac{2}{ \sqrt{2 {}^{2} \times 2} }$

$\dfrac{2}{ \sqrt{2 {}^{2} } \sqrt{2} }$

$\dfrac{2}{2 \sqrt{2} }$

$\dfrac{1}{ \sqrt{2} }$

$\dfrac{1}{ \sqrt{2} } \times \dfrac{ \sqrt{2} }{ \sqrt{2} }$

$\dfrac{1 \sqrt{2} }{ \sqrt{2} \sqrt{2} }$

$\dfrac{ \sqrt{2} }{ \sqrt{4} }$

$\dfrac{ \sqrt{2} }{2}$

D)

$\dfrac{ \sqrt{5} - \sqrt{3} }{ \sqrt{3} }$

$\dfrac{ \sqrt{5} - \sqrt{3} }{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\dfrac{( \sqrt{5} - \sqrt{3}) \sqrt{3} }{ \sqrt{3} \sqrt{3} }$

$\dfrac{ \sqrt{5} \sqrt{3} - \sqrt{3} \sqrt{3} }{ \sqrt{9} }$

$\dfrac{ \sqrt{15} - \sqrt{9} }{3}$

$\dfrac{ \sqrt{15} - 3 }{3}$

E)

$\dfrac{3 + \sqrt{3} }{ \sqrt{3} }$

$\dfrac{3 + \sqrt{3} }{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\dfrac{(3 + \sqrt{3}) \sqrt{3} }{ \sqrt{3} \sqrt{3} }$

$\dfrac{3 \sqrt{3} + \sqrt{3} \sqrt{3} }{ \sqrt{9} }$

$\dfrac{3 \sqrt{3} + \sqrt{9} }{3}$

$\dfrac{3 \sqrt{3} + 3 }{3}$

$3 \sqrt{3} + 1$

F)

$\dfrac{10}{2 - \sqrt{2} }$

$\dfrac{10}{2 - \sqrt{2} } \times \dfrac{2 + \sqrt{2} }{2 + \sqrt{2} }$

$\dfrac{10(2 + \sqrt{2}) }{(2 - \sqrt{2} ) \times (2 + \sqrt{2}) }$

$\dfrac{10(2 + \sqrt{2}) }{(2) {}^{2} - ( \sqrt{2}) {}^{2} }$

$\dfrac{10(2 + \sqrt{2}) }{4 - 2}$

$\dfrac{10(2 + \sqrt{2}) }{2}$

$5(2 + \sqrt{2} )$

$10 + 5 \sqrt{2}$

G)

$\dfrac{3}{ \sqrt{5} + \sqrt{3} }$

$\dfrac{3}{ \sqrt{5} + \sqrt{3} } \times \dfrac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} }$

$\dfrac{3( \sqrt{5} - \sqrt{3}) }{( \sqrt{5} + 3)( \sqrt{5} - 3)}$

$\dfrac{3 \sqrt{5} - 3 \sqrt{3} }{( \sqrt{5} ) {}^{2} - ( \sqrt{3} ) {}^{2} }$

$\dfrac{3 \sqrt{5} - 3 \sqrt{3} }{5 - 3}$

$\dfrac{3 \sqrt{5} - 3 \sqrt{3} }{2}$

H)

$\dfrac{ \sqrt{2} }{ \sqrt{3} - 1}$

$\dfrac{ \sqrt{2} }{ \sqrt{3} - 1 } \times \dfrac{ \sqrt{3} + 1 }{ \sqrt{3} + 1 }$

$\dfrac{ \sqrt{2} ( \sqrt{3} + 1) }{( \sqrt{3} - 1)( \sqrt{3} + 1) }$

$\dfrac{ \sqrt{2} \sqrt{3} + 1 \sqrt{2} }{( \sqrt{3}) {}^{2} - ( \sqrt{1} ) {}^{2} }$

$\dfrac{ \sqrt{6} + \sqrt{2} }{3 - 1}$

$\dfrac{ \sqrt{6} + \sqrt{2} }{2}$

I)

$\dfrac{ \sqrt{11} + 1 }{ \sqrt{11} - 1 }$

$\dfrac{ \sqrt{11} + 1 }{ \sqrt{11} - 1} \times \dfrac{ \sqrt{11} + 1 }{ \sqrt{11} + 1 }$

$\dfrac{( \sqrt{11} + 1) \times ( \sqrt{11 } + 1)}{( \sqrt{11} - 1) \times ( \sqrt{11} + 1) }$

$\dfrac{( \sqrt{11} + 1) {}^{2} }{( \sqrt{11}) {}^{2} - (1) {}^{2} }$

$\dfrac{( \sqrt{11}) {}^{2} + 2 \times \sqrt{11} \times 1 + (1) {}^{2} }{11 - 1}$

$\dfrac{11 + 2 \sqrt{11} + 1 }{10}$

$\dfrac{11 + 1 + 2 \sqrt{11} }{10}$

$\dfrac{12 + 2 \sqrt{11} }{10}$

Simplifica tudo por 2

$\dfrac{6 + \sqrt{11} }{5}$

J)

$\dfrac{7}{2 + \sqrt{5} }$

$\dfrac{7}{2 + \sqrt{5} } \times \dfrac{2 - \sqrt{5} }{2 - \sqrt{5} }$

$\dfrac{7(2 - \sqrt{5} )}{(2 + \sqrt{5} ) \times (2 - \sqrt{5}) }$

$\dfrac{7(2 - \sqrt{5}) }{(2) {}^{2} - ( \sqrt{5}) {}^{2} }$

$\dfrac{7(2 - \sqrt{5}) }{4 - 5}$

$\dfrac{7(2 - \sqrt{5} )}{ - 1}$

$- 7(2 - \sqrt{5} )$

$- 14 + 7 \sqrt{5}$